Theory of Games and Economic Behavior

PREFACE TO FIRST EDITION 10
PREFACE TO SECOND EDITION 10
PREFACE TO THIRD EDITION 12
TECHNICAL NOTE 14
CONTENTS 16
CHAPTER I FORMULATION OF THE ECONOMIC PROBLEM 26
1. The Mathematical Method in Economics 26
1.1. Introductory Remarks 26
1.2. Difficulties of the Application of the Mathematical Method 27
1.3. Necessary Limitations of the Objectives 31
1.4. Concluding Remarks 32
2. Qualitative Discussion of the Problem of Rational Behavior 33
2.1. The Problem of Rational Behavior 33
2.2. "Robinson Crusoe" Economy and Social Exchange Economy 34
2.3. The Number of Variables and the Number of Participants 37
2.4. The Case of Many Participants : Free Competition 38
2.5. The "Lausanne" Theory 40
3. The Notion of Utility 40
3.1. Preferences and Utilities 40
3.2. Principles of Measurement : Preliminaries 41
3.3. Probability and Numerical Utilities 42
3.4. Principles of Measurement : Detailed Discussion 45
3.6. Conceptual Structure of the Axiomatic Treatment of Numerical Utilities 49
3.6. The Axioms and Their Interpretation 51
3.7. General Remarks Concerning the Axioms 53
3.8. The Role of the Concept of Marginal Utility 54
4. Structure of the Theory : Solutions and Standards of Behavior 56
4.1. The Simplest Concept of a Solution for One Participant 56
4.2. Extension to All Participants 58
4.3. The Solution as a Set of Imputations 59
4.4. The Intransitive Notion of "Superiority" or "Domination" 62
4.5. The Precise Definition of a Solution 64
4.6. Interpretation of Our Definition in Terms of "Standards of Behavior" 65
4.7. Games and Social Organizations 68
4.8. Concluding Remarks 68
CHAPTER II GENERAL FORMAL DESCRIPTION OF GAMES OF STRATEGY 71
5. Introduction 71
5.1. Shift of Emphasis from Economics to Games 71
5.2. General Principles of Classification and of Procedure 71
6. The Simplified Concept of a Game 73
6.1. Explanation of the Termini Technici 73
6.2. The Elements of the Game 74
6.3. Information and Preliminarity 76
6.4. Preliminarity, Transitivity, and Signaling 76
7. The Complete Concept of a Game 80
7.1. Variability of the Characteristics of Each Move 80
7.2. The General Description 82
8. Sets and Partitions 85
8.1. Desirability of a Set-theoretical Description of a Game 85
8.2. Sets, Their Properties, and Their Graphical Representation 86
8.3 Partitions, Their Properties and Their Graphical Representation 88
8.4. Logistic Interpretation of Sets and Partitions 91
9. The Set-theoretical Description of a Game 92
9.1. The Partitions Which Describe a Game 92
9.2. Discussion of These Partitions and Their Properties 96
10. Axiomatic Formulation 98
10.1. The Axioms and Their Interpretations 98
10.2. Logistic Discussion of the Axioms 101
10.3. General Remarks Concerning the Axioms 101
10.4. Graphical Representation 102
11. Strategies and the Final Simplification of the Description of a Game 104
11.1. The Concept of a Strategy and Its Formalization 104
ll.2. The Final Simplification of the Description of a Game 106
11.3. The Role of Strategies in the Simplified Form of a Game 109
11.4. The Meaning of the Zero-sum Restriction 109
CHAPTER III ZERO-SUM TWO-PERSON GAMES: THEORY 110
12. Preliminary Survey 110
12.1. General Viewpoints 110
12.2. The One-person Game 110
12.3. Chance and Probability 112
12.4. The Next Objective 112
13. Functional Calculus 113
13.1. Basic Definitions 113
13.2. The Operations Max and Min 114
13.3. Commutativity Questions 116
13.4. The Mixed case. Saddle Points 118
13.5. Proofs o! the Main Facts 120
14. Strictly Determined Games 123
14.1. Formulation of the Problem 123
14.2. The Minorant and the Majorant Gaifces 125
14.3. Discussion of the Auxiliary Games 126
14.4. Conclusions 130
14.5. Analysis of Strict Determinateness 131
14.6. The Interchange of Players. Symmetry 134
14.7. N on -strictly Determined Games 135
14.8. Program of a Detailed Analysis of Strict Determinateness 136
15. Games with Perfect Information 137
15.1. Statement of Purpose. Induction 137
15.2. The Exact Condition (First Step) 139
15.3. The Exact Condition (Entire Induction) 141
15.4. Exact Discussion of the Inductive Step 142
15.5. Exact Discussion of the Inductive Step (Continuation) 145
15.6. The Result in the Case of Perfect Information 148
15.7. Application to Chess 149
15.8. The Alternative, Verbal Discussion 151
16. Linearity and Convexity 153
16.1. Geometrical Background 153
16.2. Vector Operations 154
16.3. The Theorem of the Supporting Hyperplanes 159
16.4. The Theorem of the Alternative for Matrices 163
17. Mixed Strategies. The Solution for All Games 168
17.1. Discussion of Two Elementary Examples 168
17.2. Generalization of This View Point 170
17.3. Justification of the Procedure As Applied to an Individual Play 171
17.4. The Minorant and the Majorant Games (For Mixed Strategies) 174
17.5. General Strict Determinateness 175
17.6 Proof of the Main Theorem 178
17.7. Comparison of the Treatments by Pure and by Mixed Strategies 180
17.8. Analysis of General Strict Determinateness 183
17.9. Further Characteristics of Good Strategies 185
17.10. Mistakes and Their Consequences. Permanent Optimality 187
17.11. The Interchange of Players. Symmetry 190
CHAPTER IV ZERO-SUM TWO-PERSON GAMES: EXAMPLES 194
18. Some Elementary Games 194
18.1. The Simplest Games 194
18.2. Detailed Quantitative Discussion of These Games 195
18.3. Qualitative Characterizations 198
18.4. Discussion of Some Specific Games (Generalized Forms of Matching Pennis) 200
18.5. Discussion of Some Slightly More Complicated Games 203
18.6. Chance and Imperfect information 207
18.7. Interpretation of This Result 210
19. Poker and Bluffing 211
19.1. Description of Poker 211
19.2. Bluffing 213
19.3. Description of Poker (Continued) 214
19.4. Exact Formulation of the Rules 215
19.6. Description of the Strategies 216
19.6. Statement of the Problem 220
19.7. Passage from the Discrete to the Continuous Problem 221
19.8. Mathematical Determination of the Solution 224
19.9. Detailed Analysis of the Solution 227
19.10. Interpretation of the Solution 229
19.11. More General Forms of Poker 232
19.12. Discrete Hands 233
19.13. m possible Bids 234
19.14. Alternate Bidding 236
19.15. Mathematical Description of All Solutions 241
19.16. Interpretation of the Solutions. Conclusions 243
CHAPTER V ZERO-SUM THREE-PERSON GAMES 245
20. Preliminary Survey 245
20.1. General Viewpoints 245
20.2. Coalitions 246
21. The Simple Majority Game of Three Persons 247
21.1. Description of the Game 247
21.2. Analysis of the Game. Necessity of "Understandings" 248
21.3. Analysis of the Game : Coalitions. The Role of Symmetry 249
22. Further Examples 250
22.1. Unsymmetric Distribution. Necessity of Compensations 250
22.2. Coalitions of Different Strength. Discussion 252
22.3. An Inequality. Formulae 254
23. The General Case 256
23.1. Exhaustive Discussion. Inessential and Essential Games 256
23.2. Complete Formulae 257
24. Discussion of an Objection 258
24.1. The Case of Perfect Information and Its Significance 258
24.2. Detailed Discussion. Necessity of Compensations between Three or More Players 260
CHAPTER VI FORMULATION OF THE GENERAL THEORY: ZERO-SUM n-PERSON GAMES 263
25. The Characteristic Function 263
25.1. Motivation and Definition 263
25.2. Discussion of the Concept 265
26.3. Fundamental Properties 266
25.4. Immediate Mathematical Consequences 267
26. Construction of a Game with a Given Characteristic Function 268
26.1. The Construction 268
26.2. Summary 270
27. Strategic Equivalence. Inessential and Essential Games 270
27.1. Strategic Equivalence. The Reduced Form 270
27.2. Inequalities. The Quantity r 273
27.3. Inessentiality and Essentiality 274
27.4. Various Criteria. Non-additive Utilities 275
27.5. The Inequalities in the Essential Case 277
27.6. Vector Operations on Characteristic Functions 278
28. Groups, Symmetry and Fairness 280
28.1. Permutations, Their Groups, and Their Effect on a Game 280
28.2. Symmetry and Fairness 283
29. Reconsideration of the Zero-sum Three-person Game 285
29.1. Qualitative Discussion 285
29.2. Quantitative Discussion 287
30. The Exact Form of the General Definitions 288
30.1. The Definitions 288
30.2. Discussion and Recapitulation 290
30.3 The Concept of Saturation 291
30.4. Three Immediate Objectives 296
31. First Consequences 297
31.1. Convexity, Flatness, and Some Criteria for Domination 297
31.2. The System of All Imputations. One -element Solutions 302
31.3. The Isomorphism Which Corresponds to Strategic Equivalence 306
32. Determination of all Solutions of the Essential Zero-sum Three-person Game 307
32.1. Formulation of the Mathematical Problem. The Graphical Method 307
32.2 Determination of ALL Solutions 310
33. Conclusions 313
33.1. The Multiplicity of Solutions. Discrimination and Its Meaning 313
33.2. Statics and Dynamics 315
CHAPTER VII ZERO-SUM FOUR-PERSON GAMES 316
34. Preliminary Survey 316
34.1. General Viewpoints 316
34.2. Formalism of the Essential Zero -sum Four-person Game 316
34.3. Permutations of the Players 319
35. Discussion of Some Special Points in the Cube Q 320
35.1 The Corner I (and V,VI, VII) 320
35.2. The Corner VIII (and II, III, IV). The Three-person Game and a "Dummy 324
35.3. Some Remarks Concerning the Interior of Q 327
36. Discussion of the Main Diagonals 329
36.1. The Part Adjacent to the Corner VIII.: Heuristic Discussion 329
36.2. The Part Adjacent to the Corner VIII. : Exact 332
36.3. Other Parts of the Main Diagonals 337
37. The Center and Its Environs 338
37.1. First Orientation Concerning the Conditions around the Center 338
37.2. The Two Alternatives and the Role of Symmetry 340
37.3. The First Alternative at the Center 341
37.4. The Second Alternative at the Center 342
37.5. Comparison of the Two Central Solutions 343
37.6. Unsymmetrical Central Solutions 344
38. A Family of Solutions for a Neighborhood of the Center 346
38.1. Transformation of the Solution Belonging to the First Alternative at the Center 346
38.2. Exact Discussion 347
38.3. Interpretation of The Solutions 353
CHAPTER VIII SOME REMARKS CONCERNING n >=5 PARTICIPANTS 355
39. The Number of Parameters in Various Classes of Games 355
39.1. The Situation for n = 3,4 355
39.2. The Situation for All n>=3 355
40. The Symmetric Five -person Game 357
40.1. Formalism of the Symmetric Five-person Game 357
40.2. The Two Extreme Cases 357
40.3. Connection between the Symmetric Five-person Game and the 1,2,3-symmetric Four-person Game 359
CHAPTER IX COMPOSITION AND DECOMPOSITION OF GAMES 364
41. Composition and Decomposition 364
41.1. Search for n-person Games for Which All Solutions Can Be Determined 364
41.2. The First Type. Composition and Decomposition 365
41.3. Exact Definitions 366
41.4. Analysis of Decomposability 368
41.5. Desirability of a Modification 370
42. Modification of the Theory 370
42.1. No Complete Abandoning of the Zero-sum Condition 370
42.2. Strategic Equivalence. Constant-sum Games 371
42.3. The Characteristic Function in the New Theory 373
42.4. Imputations, Domination, Solutions in the New Theory 375
42.5. Essentiality, Inessentiality, and Decomposability in the New Theory 376
43. The Decomposition Partition 378
43.1. Splitting Sets. Constituents 378
43.2. Properties of the System of All Splitting Sets 378
43.3. Characterization of the System of All Splitting Sets. The Decomposition Partition 379
43.4. Properties of the Decomposition Partition 382
44. Decomposable Games. Further Extension of the Theory 383
44.1. Solutions of a (Decomposable) Game and Solutions of Its Constituents 383
44.2. Composition and Decomposition of Imputations and of Sets of Imputations 384
44.3. Composition and Decomposition of Solutions. 386
44.4. Extension of the Theory. Outside Sources 388
44.5. The Excess 389
44.6. Limitations of the Excess. 391
44.7. Discussion of the New Setup 392
45. Limitations of Excess. Structure of Extended Theory 393
45.1 The Lower Limit of the Excesses 393
45.2. The Upper Limit of the Excess. Detached and Fully Detached Imputations 394
45.3 Discussion of the Two Limits 397
45.4. Detached Imputations and Various Solutions. 400
45.5. Proof of the Theorem 401
45.6. Summary and Conclusions 405
46. Determination of All Solutions in a Decomposable Game 406
46.1. Elementary Properties of Decompositions 406
46.2. Decomposition and Its Relation to the Solutions: First Results Concerning F(e ) 409
46.3. Continuation 411
46.4 Continuation 413
46.5. The Complete Result in F(e Q ) 415
46.6. The Complete Result in E(e ) 418
46.7 Graphical Representation of a Part of the Result 419
46.8. Interpretation : The Normal Zone. Heredity of Various Properties 421
46.9. Dummies 422
46.10 Imbedding a Game 423
46.11. Significance of the Normal Zone 426
46.12. First Occurrence of the Phenomenon of Transfer: n - 6 427
47. The Essential Three-person Game in the New Theory 428
47.1. Need for This Discussion 428
47.2. Preparatory Considerations 428
47.3. The Six Cases of the Discussion. Cases (I)-(III) 431
47.4. Case (IV) : First Part 432
47.5 Case (IV) : Second Part 434
47.6. Case (V) 438
47.7 Case (VI) 440
47.8. Interpretation of the Result: The Curves (One Dimensional Parts) in the Solution 441
47.9. Continuation : The Areas (Two-dimensional Parts) in the Solution 443
CHAPTER X SIMPLE GAMES 445
48. Winning and Losing Coalitions and Games Where They Occur 445
48.1. The Second Type of 41.1. Decision by Coalitions 445
48.2. Winning and Losing Coalitions 446
49. Characterization of the Simple Games 448
49.1. General Concepts of Winning and Losing Coalitions 448
49.2. The Special Role of One-element Sets 450
49.3. Characterization of the Systems W, L of Actual Games 451
49.4. Exact Definition of Simplicity 453
49.6. Some Elementary Properties of Simplicity 453
49.6. Simple Games and Their W, L. The Minimal Winning Coalitions : W^m 454
49.7. The Solutions of Simple Games 455
50. The Majority Games and the Main Solution 456
50.1. Examples of Simple Games : The Majority Games 456
50.2. Homogeneity 458
50.3. A More Direct Use of the Concept of Imputation in Forming Solutions 460
50.4. Discussion of This Direct Approach 461
50.5. Connection with the General Theory. Exact Formulation 463
60.6. Reformulation of the Result 465
50.7. Interpretation of the Result 467
50.8. Connection with the Homogeneous Majority Games 468
51. Methods for the Enumeration of All Simple Games 470
51.1. Preliminary Remarks 470
51.2. The Saturation Method : Enumeration by Means of W 471
51.3. Reasons for Passing from W to W^m. Difficulties of Using W^m 473
51.4. Changed Approach : Enumeration by Means of W^m 475
51.5. Simplicity and Decomposition 477
51.6. Inessentiality, Simplicity and Composition. Treatment of the Excess 479
51.7. A Criterion of Decomposability in Terms of W^m 480
52. the Simple Games for Small n 482
52.1. Program: n 1, 2 Play No Role. Disposal of n = 3 482
52.3. Decomposability of the Cases 484
52.4. The Simple Games Other than [1, , 1, I - 2]* (with Dummies) 486
52.5. Disposal of n = 4, 5 487
53. The New Possibilities of Simple Games for n>=6 488
53.1. The Regularities Observed for n < 6 488
5S.2. The Six Main Counter-examples (for n 6, 7) 489
54. Determination of All Solutions in Suitable Games 495
54.1 Reasons to Consider Solutions than the Main Solution in Simple Games 495
54.2. Enumeration of Those Games for Which All Solutions Are Known 496
54.3. Reasons to Consider the Simple Game [1, - , 1, n 2]* 497
55. The Simple Game [1, , 1, n - 2] h 498
55.1. Preliminary Remarks 498
55.2. Domination. The Chief Player. Cases (I) and (II) 498
55.3. Disposal of Case (I) 500
55.4 Case (III): Determination of V 503
55.5. Case (II) : Determination of V 506
55.6. Case (II) : a and S+ 509
55.7. Cases (II') and (II"). Disposal of Case (II') 510
55.8. Case (II") : a and V. Domination 512
55.9 Case (II): Determination of V 513
55.10. Disposal of Case (II") 519
55.11.Reformulation of the Complete Result 522
55.12. Interpretation of the Result 524
CHAPTER XI GENERAL NON-ZERO-SUM GAMES 529
56. Extension of the Theory 529
56.1. Formulation of the Problem 529
56.2. The Fictitious Player. The Zero-sum Extension 530
56.3. Questions Concerning the Character of P 531
56.4. Limitations of the Use of f 533
56.5. The Two Possible Procedures 535
56.6. The Discriminatory Solutions 536
56.7. Alternative Possibilities 537
56.8. The New Setup 539
56.9. Reconsideration of the Case Where T is a Zero-sum Game 541
56.10. Analysis of the Concept of Domination 545
56.11. Rigorous Discussion 548
56.12 The New Definition of a Solution 551
57. The Characteristic Function and Related Topics 552
57.1. The Characteristic Function : The Extended and the Restricted Forms 552
57.2. Fundamental Properties 553
57.3. Determination of All Characteristic Functions 555
57.4. Removable Sets of Players 558
57.5. Strategic Equivalence. Zero-sum and Constant-sum Games 560
58. Interpretation of the Characteristic Function 563
58.1. Analysis of the Definition 563
58.2 The Desire to make a Gain vs That to inflict a loss 564
58.3. Discussion 566
59. General Considerations 567
59.1. Discussion of the Program 567
59.2. The Reduced Forms. The Inequalities 568
59.3. Various Topics 571
60. The Solutions of All General Games with n^3 573
60.1. The Case n-1 573
60.2 The Case n=2 574
60.3 The case n=3 575
61. Economic Interpretation of the Results for n = 1,2 580
61.1. The Case n-1 580
61.2. The Case n = 2. The Two-person Market 580
61.3. Discussion of the Two-person Market and Its Characteristic Function 582
61.4. Justification of the Standpoint of 68. 584
61.6. Divisible Goods. The "Marginal Pairs" 585
61.6. The Price. Discussion 587
62. Economic Interpretation of the Results for n = 3 : Special Case 589
62.1. The Case n 3, Special Case. The Three-person Market 589
62.2. Preliminary Discussion 591
62.3. The Solutions : First Subcase 591
62.4. The Solutions : General Form 594
62.6. Algebraical Form of the Result 595
62.6. Discussion 596
63. Economic Interpretation of the Results for n = 3 : General Case 598
63.1. Divisible Goods 598
63.2. Analysis of the Inequalities 600
63.3. Preliminary Discussion 602
63.4. The Solutions 602
63.6. Algebraic Form of the Result 605
68.6. Discussion 606
64. The General Market 608
64.1. Formulation of the Problem 608
64.2. Some Special Properties. Monopoly and Monopsony 609
CHAPTER XII EXTENSIONS OF THE CONCEPTS OF DOMINATION AND SOLUTION 612
65. The Extension. Special Cases 612
66.1. Formulation of the Problem 612
66.2. General Remarks 613
66.3. Orderings, Transitivity, Acyclicity 614
65.4. The Solutions : For a Symmetric Relation. For a Complete Ordering 616
66.5. The Solutions : For a Partial Ordering 617
66.6. Acyclicity and Strict Acyclicity 619
65.7. The Solutions : For an Acyclic Relation 622
66.8. Uniqueness of the Solutions, Acyclicity and Strict Acyclicity 625
66.9. Application to Games : Discreteness and Continuity 627
66. Generalization of the Concept of Utility 628
66.1. The Generalization. The Two Phases of the Theoretical Treatment 628
66.2. Discussion of the First Phase 629
66.3. Discussion of the Second Phase 631
66.4. Desirability of Unifying the Two Phases 632
67. Discussion of an Example 633
67.1. Description of the Example 633
67.2. The Solution and Its Interpretation 636
67.3. Generalization : Different Discrete Utility Scales 639
67.4. Conclusions Concerning Bargaining 641
APPENDIX. THE AXIOMATIC TREATMENT OF UTILITY 642
A.I. Formulation of the Problem 642
A.2. Derivation from the Axioms 643
A.3. Concluding Remarks 653
INDEX OF SUBJECTS 660